where \(\beta (z,w)\)  stands for the Bergman distance. Since \(\beta \)  is Möbius invariant (see, e.g.,[1]}), it follows that one can find such \(z_1, z_2\in K\)  that for all \(z\in U\)
\(|f(\varphi _w(z))|\le \Vert f\Vert _{\mathcal {B}}\beta (0,\varphi _w(z))+|f(-z)|= \Vert f\Vert _{\mathcal {B}}\beta (\varphi _w(w),\varphi _w(z))+|f(-z)|=\) 
\(\Vert f\Vert _{\mathcal {B}}\beta (w,z)+|f(-z)|\le \Vert f\Vert _{\mathcal {B}}(\beta (w,0)+\beta (0,z))+|f(-z)|\le \) 
\(\Vert f\Vert _{\mathcal {B}}\left(\frac{1}{2}\log \frac{1+|w|}{1-|w|}+\beta (0,z_1)\right)+|f(-z_2)|.\) 
